ROW ELEMENTS AND THE PERFORMANCE OF THEORY

COMPUTATIONAL STUDIES OF REACTIONS INVOLVING

1sT, 2ND

AND

3Ro

ROW ELEMENTS AND THE PERFORMANCE OF THEORY

BY

MOHAMMAD SHAHIDUL ISLAM

A DISSERTATION

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR TH E DEGREE OF DOCTOR OF PHILOSOPHY

ST. JOHN'S

DEPARTMENT OF CHEMISTRY

MEMORIAL UNIVERSITY OF NEWFOUNDLAND

DECEMBER, 2007

NEWFOUNDLAND CANADA

I

I

I I

I I I I

I

Abstract

This thesis has involved detailed computational studies of the mechanisms of several chemical reactions involving first, second and third row elements. Geometries of the relevant molecules were optimized at the HF. MP2 and B3 L YP levels using the 6- 31 G(d), 6-31 +G(d). 6-31 G(d,p) and aug-cc-pVDZ basis sets. Gaussian-n theories such as G3MP2, G3MP283 and G3B3 were also used, as they are expected to adequately reproduce experimental data. The complete reaction pathways for all the mechanisms have been verified using intrinsic reaction coordinate (IRC) analysis.

The reactions of SiH₃X (X = H, Cl, Br, I) with HCN were investigated and three different mechanisms were obtained. One of the mechanisms involves HX elimination by a one-step pathway producing SiH3CN. The second mechanism consists of H2 elimination, producing Sii-hXCN via a one-step pathway or three multiple-step pathways.

The third mechanism involves dissociation of SiH3X to various products, which can then react with HC .

We have found for the first time that the mechanism of the addition of bromine to alkenes involves reaction with two bromine molecules in non-polar aprotic solvents, while in polar protic solvents the mechanism involves reaction with a single bromine molecule mediated by a solvent molecule. For both cases, the calculated activation energies were found to be in excellent agreement with experiment. We proposed a kinetic

expression that accounts for the difference between bromination of alkenes in protic and aprotic solvents. We also found that bromination of adamantylideneadamantane should occur spontaneously in the gas phase as well as in some solvents with no reaction barrier.

We have found that for third row elements the BC6-31 G basis set which is widely used as a 6-31 G basis set in most of the commercial quantum chemistry packages does not meet the definition of the standard 6-31 G basis set. A comparative study of the performance of the standard 6-31 G and Binning-Curtiss (BC6-31 G) basis sets for third row elements, Ga, Ge, As, Se, and Br, was carried out. Frequencies and thermodynamic values obtained by using the standard 6-3 I G basis set are better than those obtained using the BC6-31 G basis set when compared to experiment and G3MP2. We recommend that the standard 6-31 G basis set be used for calculations involving 3rd row elements.

The kinetic isotope effects (KIEs), a major experimental tool to determine the transition state (TS) structure, have been used to characterize the transition state structure of SN2 reactions. Chlorine leaving group k³⁵ /k^37, nucleophile carbon k ¹¹/k ¹⁴ and secondary a-deuterium [(kH/kD)a] kinetic isotope effects (KIEs) have been calculated for the SN2 reactions between para-substituted benzyl chlorides and cyanide ion and compared to the experimental results to determine whether these isotope effects can be used to determine the substituent effect on the structure of the transition state. It was found that both leaving group and nucleophile Kl E vary with the TS structure. However, a correct and measurable substituent effect on leaving group KIEs will only be found for

a very reactant-like or for a very product-like TS. The substituent effect on nucleophile K!Es will only be found when the u-C11 bond formation in the TS is well advanced i.e., in a product-like SN2 TS.

ucleophile carbon k ¹¹ /k ¹⁴ and secondary a-deuterium [(k1 /ko)a] kinetic i 'Otope effects (K!Es) were also calculated for the S 2 reactions between tetrabutylammonium cyanide and ethyl iodide, bromide, chloride and tosylate and compared to the experimental results to determine whether these isotope effects can be used to determine the structure of the SN2 transition states. The results showed that the nucleophile carbon k¹¹/k¹⁴ KIEs can be used to determine the transition state structure in different reactions and the results suggest that changing to a poorer leaving group leads to a tighter transition state. The magnitude of the experimental secondary a-deuterium KIE is related to the nucleophile - leaving group distance in the S 2 transition state (Rrs) for reactions with a halogen leaving group. However, the calculated and experimental a-deuterium KIEs show opposite trends with leaving group ability.

In conclusion, the results of my doctoral research have greatly increased our knowledge of the mechanisms, transition state structures and the thermodynamic

. f . . I . I st 2^nd d 3^rd I

properties o reactiOns mvo vmg , an row e ements.

Acknowledgements

All praises to the Almighty, the Most Gracious and the Most Benevolent God (J\llah) for his infinite mercy bestowed on me in completing this great task within the stipulated time and fulfilling my life dream.

In the eve of submitting this dissertation, I cannot but express my deepest sense of gratitude and indebtedness to my respected teacher and supervisor Prof Raymond A.

Poirier, Department of Chemistry, Memorial University, for hi kind advice, indispensable guidance and thoughtful suggestions throughout the research work. I must confess that without his continuous inspiration and constant supervision, it would have been difficult to complete and present this dissertation in such a form.

I would also like to thank Prof. Kenneth C. Westaway, Department of Chemistry and Biochemistry, Laurentian University for his valuable advice and various discussions during our study on kinetic isotope effect (KIE). I would also like to thank Prof. Olle Matsson, Uppsala University, Sweden for providing us experimental KIE results. Special thanks to Dr. Travis Fridgen for his valuable advice and suggestions.

I would like to thank my supervisory committee, Dr. Robert Davis and Dr.

Christopher Flinn for their valuable suggestions and support in different ways throughout my research work and for proof reading the early version of this thesis.

I would also like to acknowledge each member of our vibrant computational chemistry group at Memorial University of Newfoundland (MUN) for cordial help valuable advice, and continuous inspiration throughout the progress of this thesis work. I would like to give especial thanks to my friends and fellow colleagues, Joshua Hollett, Kushan Saputantri, Mansour Almatarneh, Aisha El-Sherbiny, Csaba Szakacs, Eva Simon, and Tammy Gosse for their sincere co-operation, many fruitful discussions and unconditional support.

I would also like to acknowledge the Department of Chemistry and the School of Graduate Studies at MUN and NSERC for financial support during my PhD program. I would also like to thank ACEnet for computer time and various agencies that have funded my conference trips.

At last, but not the least, I acknowledge my parents: my father, Prof. Md. Nurul Islam, who revealed the world of logic, and my mother, Mrs. Jahanara Islam, who revealed the world of life. My sincere gratitude is extended to them for their infinite contribution, prayer, and encouragement throughout my life. I am very grateful to my loving wife Thasin Islam for her help, inspiration, and encouragement during the period of my study. I am also grateful to my father and mother in law, my sisters Mrs. Fauzia Nur and Dr. Laila Nur, brothers Md. Zahedul Islam and Md. Kamrul Islam and brother in laws Golam Jilani and Jumar Islam for their constant inspiration, prayer, and encouragement. I would also like to extend my thanks to all my well-wishers.

DEDICATED TO

MY BELOVED PARENTS PROF.NURUL ISLAM

&

MRS. JAHANARA ISLAM

AND

MY LOVELY WIFE

THASIN JACKIE ISLAM

Table of Contents

Abstract

Acknowledgements Dedication

Table of Contents List of Tables List of Figures List of Schemes

List of Abbreviations and Symbols List of Publications from This Thesis

Chapter 1 Introduction

1.1 Overall Goals and Objectives I .2 Background

1.2.1 Transition State Theory 1.2.2 Kinetic Isotope Effect

1.2.3 Origin of Kinetic Isotope Effect 1.2.4 Calculation of Kinetic Isotope effect 1.2.5 The Molecular Hamiltonian

1.2.6 Born-Oppenheimer Approximation 1.2.7 The Hartree-Fock Approximation

11

VII

Vlll

XV

XXIII

XXVIII

XXIX

XXX Ill

5 5

7 9 10

13 14 15

1.2.8 The Basis Set Expansions 1.2.9 Post Hartree-Fock Methods

1.2.1 0 Moller-Piesset Perturbation Theory 1.2.11 Configuration Interaction

1.2.12 Density Functional Theory (DFT) 1.2.13 G3B3, G3MP2 and G3MP2B3 Theories 1.2.14 Solvation Models

1.2.15 The Onsager Model

1.2.4 Polarizable Continuum Model 1.3 References

Chapter 2 Computational Study of the Reactions of SiH3X (X

=

H, Cl, Br, I) with HCN

17 19 19 20 21 23 25 26 27 27

2.1 Introduction 37

2.2 Method 40

2.3 Results and Discussion 41

2.3.1 Activation energies and free energies of activation for the 41 reaction of SiH3X with HCN

2.3.1.1 Reaction ofSiH₃X and HCN (Pathway A) 41 2.3.1.2 Reaction of SiH₃X and 1-ICN (H2 elimination) 43 2.3.1.3 Decomposition of SiH3X and reaction with HCN 46 2.3.1.4 Summary of overall reaction mechanisms 48

investigated

2.3.2 Thermodynamic results for the reaction of Siii3X with HCN 49 2.3.2.1 Thermodynamics of HX elimination 49

2.3.2.2 Thermodynamics or Hz elimination 50

2.3.2.3 Thermodynamics for the thermal decomposition of 50 Sil-I)X and SiHX + HC

2.3.3 Performance of Theory/Basis set 2.3.4 Exploring Heats of Formation ^(6Hr) 2. 4 Conclusions

2.5 References

Chapter 3 New insights into the bromination reaction for a series of alkenes- A computational study

51 53 55 57

3. I Introduction 90

3 .2 Method 93

3.3 Results and Discussions 94

3.3.1 Potential energy surfaces for the reaction of alkenes with Br2 95 3.3.1.1 Perpendicular attack of Br2 to C=C: Pathway A 95 3.3.1.2 Sidewise attack of Br2 to C=C: Pathway B 96 3.3 .2 Potential energy surfaces for the reaction of ethene with 2Br2 99

3.3.2.1 Pathway C I 00

3.3.2.2 Pathway D 101

3.3.2.3 Pathway E I 02

3.3.2.4 Pathway F I OJ

3.3.3 Summary of overall reaction mechanisms investigated I 05

3.3.4 Comparison with Experiment I 05

3 .3.5 Thermodynamic results for the reaction of alkenes with Br₂ I 09

3.3.6 Performance of Theory/Basis set II 0

3.3.7 Exploring Heats of Formation (~Hr) 112

3.4 Conclusions 113

3.5 References 114

Chapter 4 The addition reaction of adamantylideneadamantane with Br₂ and 2Br₂ - A Computational Study

4.1 Introduction 149

4.2 Method 152

4.3 Results and Discussions 153

4.3.1 Potential energy surface for the reaction of Ad=Ad + Br2 : 153 Pathway A

4.3.2 Potential energy surfaces for the reaction of Ad=Ad with 2Br₂ 155

4.3.2.1 Pathway B 155

4.3.2.2 Pathway C 156

4.3.2.3 Pathway D 157

3.3.3 Relative Stabilities 158

4.4. Conclusions 160

4.5 References 161

Chapter 5 A comparison of the Standard 6-31 G and Binning-Curtiss Basis Sets for Third Row Elements

4.1 Introduction 182

4.2 Method 184

4.3 Results and Discussions 185

5.3.1. Geometries of molecules containing Yd row elements 186 5.3. 2. Frequencies of molecules containing 3rd row elements 187 5.3.3.Thermodynamic properties for the . .

1sogync reactions 189 involving 3rd row elements

5.3.4. Exploring Heats of Formation (6Hr) 193

5.4. Conclusions 194

5.5 References 196

Chapter 6 A new insight into using chlorine leaving group and nucleophile carbon kinetic isotope effects to determine substituent effects on the structure of SN2 transition states

6.1 Introduction 227

6.2 Method 229

6.3 Results and Discussion 230

6.4 Conclusions 247

6.5 References

Chapter 7 Can incoming nucleophile carbon kinetic isotope effects be used to determine the transition state structure for different SN2 reactions?

249

7.1 Introduction 267

7.2 Method 269

7.3 Results and Discussion 270

7.3.1. Experimental Nucleophile Carbon k¹¹/k¹⁴ KIEs 271 7.3.2 Experimental Secondary a-Deuterium KIEs 272 7.3.3 The effect of changing the leaving group on the structure of 274

the SN2 transition state

7.3.4 Experimental vs calculated nucleophile k¹¹/k¹⁴ carbon KIE 277 7.3.5 Experimental vs calculated (kH/ko)u KIEs 278

7.3.6 Effect of Solvent 279

7.3.7 The implications ofthe theoretical study on using nucleophilc 281 carbon KIEs to determine the transition states of SN2

reactions

7.4 Conclusions 7. 5 References

Chapter 8 Concluding Remarks

283 284

298

Appendix A Appendix B Appendix C Appendix D

309 311 317 324

Chapter 2 Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

Table 2.7

List of Tables

Activation energies and free energies of activation for the reaction of SiH3X (X=H, Cl, Br, I) with HCN (HX elimination) and the isomerization of SiH3CN/HX (in kJ mor') at 298.15K (Pathway A).

Activation energies and free energies of activation for the reaction of SiH3X (X=H, Cl, Br, I) with HCN (in kJ mol"¹⁾ at 298.15K (1-12

elimination: Pathways B I, B* and 82).

Activation energies and free energies of activation for the reaction of SiH3X (X=H, Cl, Br, I) with HCN (in kJ mol"¹⁾ at 298.15K (112

elimination: Pathways 83 and B4)

Activation energies, activation enthalpies and free energ1es of activation for the thermal decomposition reaction of SiH₃X (X=H, Cl, Br, I) (in kJ mor') at 298.15K (Figure 9).

Activation energies, enthalpies and free energies of activation for the reaction of SiHX (X=H, Cl, Br, I) with HCN (in kJ mor') at 298.15K (Figure I 0).

Thermodynamic properties for the reaction of SiH₃X (X=H, Cl, Br. I) with HCN (in kJ mol"¹) at 298.15K (HX elimination reaction).

Relative stabilities for the isomerization of Sil-hXC /Sii--{zXNC(X=I-L Cl, Br, I) (in kJ mor¹) at 298.15K.

61

63

65

67

69

70

72

Table 2.8

Table 2.9

Thermodynamic properties for the reaction of SiH3X (X=Cl, Br, I) with HCN (H2 elimination reaction) (in kJ mor¹⁾ at 298.15K.

Thermodynamic properties for the thermal decomposition reaction of SiH3X (X=H, Cl, Br, I) (in kJ mor') at 298.15K.

74

76

Table 2.10 Thermodynamic properties for the reaction of SiHX (X=H, Cl, Br, I) 78 with HCN (in kJ mor¹) at 298.15K.

Table 2.11 Heat of formation (6Hr) (in kJ mor') at 298.15 K. 79

Chapter 3 Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Activation energies, free energies and enthalpies of activation (kJ mor 119

1) at 298.15K for the reaction of CH2=CI-h, CH3-CH2=CH2, (CH3)2CH=CH2, CH2=CHF, CH2=CHCl, (E)-CHF=CHf and (E)- CHCl=CHCI with Br2 (Pathway A).

Activation energies, free energies and enthal pies of activation (kJ mor 121

1) at 298.15K for the reaction of CH₂=CH2, Cli3-CH₂=CH_2, (CH3)zCH=CHz, CH2=CHF and CHz=CHCI with Br2 (Pathway B).

Activation energies, free energies and enthalpies of activation (k.J mor 125

1) at 298.15K for the reaction of (E)-CHF=CHF and (E)-CHCI=CHCI with Br2 (Pathway B).

Activation energies, free energies and enthalpies of activation (kJ mor 126

1) at 298.15K for the reaction ofCI-[z=CH2 and 2Br2 (Pathway C).

Activation energies, free energies and enthalpies of activation (k.l mor 127

Table 3.6

Table 3.7

Table 3.8

Table 3.9

1) at 298.l5K for the reaction of CH2=CI-h and 2Br2 (Pathway D).

Activation energies, free energies and cnthalpies of activation (kJ mol" 128

1) at 298.l5K for the reaction ofCL-12=CH2 and 2Br2 (Pathway E).

Activation energies, free energies and enthalpies of activation (k.J mol" 129

1) at 298.15K for the reaction of CH2=CH2 and 2Br2 (Pathway F).

The calculated overall free energies of activation (k.J mo1"¹⁾ at 130 298.15K for the reaction of CH2=Cl-12 and 2Br2 m solution at

B3L YP/BC6-31 G(d) (Pathway F) and experimental values.

Thermodynamic properties (kJ mol"¹⁾ at 298.15K for the reaction of 131 CH2=Cl l2, CH3-CH2=CH2, (CH3)2CH=Cl-h CI--12=CHF, CH2=CHCI, (E)-CHF=CHF and (E)-CHCI=CIICI with Br2.

Table 3.10 Heats offormation, ~Hr. (kJ mol"¹⁾ at 298.15K. 134

Chapter 4 Table 4.1

Table 4.2

Table 4.3

Activation energies, free energies and enthalpies of activation (kJ mor 165

1) at 298.l5K for the reaction of adamantalydin adamantane with Br2

(Pathway A).

Activation energies, free energies and enthalpies of activation (kJ mor 166

1) at 298.15K for the reaction of adamantalydineadamantane with 2Br2

(Pathway B).

Activation energies, free energies and enthalpies of activation (kJ mor 167

1) at 298.l5K for the reaction of adamantalydineadamantanc with 2Br2

(Pathway C) in the gas phase.

Table 4.4 Activation energies, free energies and enthalpics of activation (kJ mor 168

1) at 298.15K for the reaction of adamantalydineadamantane with 2Br₂ (Pathway D).

Table 4.5 Relative Stabilities (kJ mor¹) at 298.15 K. 169

Chapter 5

Table 5.1 Optimized and experimental structural parameters for compounds 20 I containing 3rd row elements (bond lengths in

A

and angles in degrees).

Table 5.2 Mean Absolute Deviations for bond lengths and angles (bond lengths 205 in

A

and angles in degrees).

Table 5.3 Calculated and experimental frequencies (in cm-¹⁾ for compounds 206 . . 3rd I

contammg row e ements.

Table 5.4 Mean Absolute Deviations for frequencies (in cm-¹ ). 212 Table 5.5 Thermodynamic properties for the reactions {I) and (2) (in kJ mor¹⁾ at 213

298.15K.

Table 5.6 Thermodynamic properties for the reactions (3) and (4) (in kJ mor¹⁾ at 216 298.15K.

Table 5.7 Thermodynamic properties for the reaction (5) (in kJ mor¹) at 217 298.15K.

Table 5.8 Mean Absolute deviations for the enthalpies of reaction involving I ⁵¹ 218 and 3rd row elements, reaction (I), and 1 ^5', 2nd and 3rd row elements,

Table 5.9

Chapter 6 Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

Table 6.7

reaction (2) and (5), (in kJ mol¹).

IIeats of formation (L\Hr) (in kJ mor¹⁾ at 298.15K. 219

The experimental secondary a-deuterium KlE for the SN2 reactions 254 between cyanide ion and benzyl- and para-chlorobenzyl chloride at

0°C in THF.

The secondary a-deuterium (kH/kD)a KIEs for the SN2 reactions 255 between cyanide ion and five para-substituted benzyl chlorides at

25°C using three different levels of theory.

The experimental chlorine leaving group KIEs for the SN2 reactions 256 between cyanide ion and four para-substituted benzyl chlorides at

20°C in THF.

The calculated Ca-Cl bond length in five para-substituted benzyl 257 chloride at 25°C using three different levels of theory.

The calculated Ca- I and NC-Ccl transition-state bond length for the 258 SN2 reactions between cyanide ion and five para-substituted benzyl

chlorides at 25°C using three different levels of theory.

The chlorine (k³⁵/k³⁷⁾ leaving group KIEs for the SN2 reactions 259 between cyanide ion and five para-substituted benzyl chlorides at

25°C using lour different levels of theory and un-sealed frequencies.

The experimental chlorine (k³⁵/k³⁷⁾ leaving group KIEsa. ¹¹ for the N2 260

Table 6.8

Table 6.9

reactions of para-substituted benzyl chlorides with vanous nucleophiles at 20°C.

The experimental nucleophile carbon k ¹¹/k ¹⁴ KIEs for the S 2 261 reactions between cyanide ion and three para-substituted benzyl

chlorides at 20°C.

The nucleophile carbon k¹¹/k ¹⁴ KIEs for the SN2 reactions between 262 cyanide ion and five para-substituted benzyl chlorides at 25°C using

three different levels of theory.

Table 6.10 The individual contributions to the chlorine leaving group KIEs for 263

Table 6.11

the SN2 reactions between cyanide ion and five para-substituted benzyl chlorides at 25°C in the gas phase at the 83 L YP/aug-cc-p VDZ level of theory.

Substituent effects on the transition structures and chlorine leaving 264 group KIEs for different SN2 reactions with a series of para-

substituted benzyl chlorides at 25°C using the B3L YP/aug-cc-pVDZ level oftheory.

Table 6.12 The individual contributions to the calculated nucleophile carbon 265 (k¹¹/k¹⁴⁾ KIEs tor the S 2 reactions between cyanide ion and five

para-substituted benzyl chlorides at 25°C in the gas phase at the B3L YP/aug-cc-pVDZ level of theory.

Table 6.13 Substituent effects on the transition structures and nucleophile carbon 266 k¹¹/k¹⁴ and nitrogen k¹⁴/k¹⁵ KIEs for the SN2 reactions between

Chapter

7

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 7.6

cyanide ion and ammonia with a series of para-substituted benzyl chlorides, respectively, at 25°C using the B3L YP/aug-cc-pYDZ level of theory.

The rate constants and the nuclcophile carbon k¹¹/k¹~ K!Es for the S 2 288 reactions between cyanide ion and ethyl chloride, bromide, iodide and

tosylate in anhydrous DMSO at 20°C.

The secondary a-deuterium KlEs for the SN2 reaction between 289 cyanide ion and ethyl chloride, bromide, iodide and tosylate in

anhydrous DMSO at 20°C.

The calculated C0-LG and NC -Cu transition-state bond lengths for the 290 SN2 reactions between cyanide ion and ethyl iodide, ethyl bromide,

ethyl tosylate and ethyl chloride at 25°C.

The nucleophile carbon (k¹¹/k¹⁴⁾ KIEs for the SN2 reactions between 291 cyanide ion and ethyl iodide, ethyl bromide, ethyl tosylate and ethyl

chloride at 25°C ·using three different levels of theory.

The secondary a-deuterium [(kH/ko)a] KIEs for the SN2 reactions 292 between cyanide ion and ethyl iodide, ethyl bromide, ethyl tosylate

and ethyl chloride at 25°C using three different levels of theory

Free energies of activation for the S 2 reactions between cyanide ion 293 and ethyl chloride, bromide, iodide and tosylate at 20°C

Table 7.7

Table 7.8

Table 7.9

The calculated C1.-LG and NC-Ca transition-state bond lengths for the 294 SN2 reactions between cyanide ion and ethyl iodide, ethyl bromide,

ethyl tosylate and ethyl chloride in DMSO at 30°C.

The nucleophile carbon (k¹¹/k¹~) KfEs for the SN2 reactions between 295 cyanide ion and ethyl iodide, ethyl bromide, ethyl tosylate and ethyl

chloride in DMSO at 30°C using three ditTerent levels of theory.

The secondary a-deuterium (kr/ko)u KIEs for the S 2 reactions 296 between cyanide ion and ethyl iodide, ethyl bromide, ethyl tosylate

and ethyl chloride in DMSO at 30°C using three different levels of theory

Table 7.10 The individual contributions to the calculated nucleophile carbon 297 (k ¹¹ /k ¹~) KIEs for the SN2 reactions between cyanide ion and ethyl

iodide, ethyl bromide, ethyl tosylate and ethyl chloride at 25°C in the gas phase at the B3L YP/aug-cc-pVDZ level of theory.

Chapter 1 Figure 1.1

Figure 1.2 Figure 1.3 Figure 1.4

Chapter 2 Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

List of Figures

Transition state structure for the SN2 reaction between an ethyl substrate and a nucleophile showing different kinetic isotop~ effects.

Here, D represents deuterium.

Origin of k₁-/k₀ in an exothermic reaction (early TS) Origin of low k1 /ko in an endothermic reaction (late TS)

Origin of large k11/ko in a thermoneutral reaction (symmetric T )

Mechanism for the reaction of SiH₃X + HCN (Pathway A). Only X=

Cl is shown here, similar structures are found for X=Br and I.

Reaction pathway for the reaction of SiH₃X + HCN (Pathway ) at G3MP2 level of theory (see Figure I).

Mechanism for the reaction of SiH₃X + HCN (H2 elimination). Only X= Cl is shown here, similar structures are found for X=H, Brand I.

Mechanism for the reaction of SiH3X + HCN forming the intermediate I I s• (H2 elimination). Only X=Cl is shown here, similar structures are found for X=H, Brand I.

Reaction pathway 81 and 8* for the reaction of SiH3X HCN (H2

elimination) at G3MP2 level of theory (see Figure 3).

Reaction pathway 82 for the reaction of SiH3X + HC (H2

33

34

35 36

80

81

82

83

84

85

Figure 2.7

Figure 2.8

Figure 2.9

elimination) at G3MP2 level of theory (see Figure 4).

Reaction pathway B3 for the reaction of SiH₃X + IICN (1--h elimination) at G3 MP2 level of theory (sec figure 4 ).

Reaction pathway B4 for the reaction of Sil-13X + IICN (1-h elimination) at G3MP2 level of theory (see Figure 4).

Mechanism for thermal decomposition of Sil-hX. Only X=H and Cl are shown here, similar structures to X=Cl are found for X=Br and I.

86

87

88

Figure 2.10 Mechanism for the addition reaction of SiHX + HCN. Only X=CI are 89

Chapter 3 Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

shown here, similar structures are found for X=H, Brand I.

Pathway and barrier for the reaction of CH2=CH2 + Br1 (Pathway A) at MP2/BC6-31 G(d) level of theory. Although the lRC leads to lA no optimized structure was found.

Mechanism for the reaction of CH2=CH2 + Br2 (Pathway B). Similar structures are found for the bromination of CI-I)-CH₂=CI {z, (CH3)2CH=CH2, CH2=CHF and CH2=CHCI.

Pathway for the reaction of H2=CH2 Br2 (Pathway B) at MP2/BC6-31 G(d) level of theory (see Figure 2 for structures).

Mechanism for the reaction of (E)-CHF=CHF + Br2 (Pathway B).

Similar structure arc found for the bromination or (E)-CH l=CIICI.

Mechanism for the reaction of CII₂=CI 1₂ +2Br2 (Pathway C).

135

136

137

138

139

Figure 3.6 Pathway for the reaction of CH2=CH2 + 2Br2 (Pathway C) at 140 G3MP2B3 level of theory (see Figure 5 for structures).

Figure 3.7 Mechanism for the reaction of CH₂=CH2 +2Br₂ (Pathway D). 141 Figure 3.8 Pathway for the reaction of CH2=CH₂ + 2Br₂ (Pathway D) at 142

G3MP2B3 level of theory (see Figure 7 for structures).

Figure 3.9 Mechanism for the reaction of CH2^{=CH~ + 2Br}2 (Pathway ^E). 143 Figure 3.10 Pathway for the reaction of CH₂=CH2 + 2Br₂ (Pathway E) at 144

MP2/G3MP21arge//HF/6-31 G(d) level of theory (see Figure 9 for structures).

Figure 3.11 Mechanism for the reaction ofCH~=CH

2

^{+ 2Br2} (Pathway F). 145 Figure 3.12 Pathway for the reaction of CH

2

^{=CH~ + 2Br}2 (Pathway F) at 146

G3MP2B3 level of theory (see Figure II for structures). For TS2F, 12F and 13F the G3MP2B3 energies are calculated using HF/6-31 G(d) by optimized geometries.

Figure 3.13 Mechanism for the reaction of Br₂ + Br2 ^~ Br₄^. 14 7 Figure 3.14 Mechanism for the reaction of CH₂=CH₂ + Br₂ mediated by a 148

CH30H molecule (Pathway G).

Chapter 4

Figure 4. I Expected structure of the adamantylideneadamantane bromonium 173

IOn.

Figure 4.2 Mechanism for the reaction of Ad=Ad + Br2 (Pathway A). 174

Figure 4.3 Reaction pathway for the reaction of Ad=J\d + Br₂ (Pathway A) at 175 HF/6-31 G(d) and B3L YP/6-31 G(d) level of theory (see Figure 2 for

structures).

Figure 4.4 Mechanism for the reaction of Ad=Ad + 2Br₂ (Pathway B). 176 Figure 4.5 Reaction pathway for the reaction of Ad=Ad + 2Br₂ (Pathway B) at 177

HF/6-31 G(d) and 83L YP/6-31 G(d) level of theory (see Figure 4 for structures).

Figure 4.6 Mechanism for the reaction of Ad=Ad + 2Br₂ (Pathway C). 178 Figure 4.7 Reaction pathway for the reaction of Ad=Ad + 2Br₂ (Pathway C) at 179

HF/6-31G(d) level of theory. For Rc and TSc BJLYP/6- 31 G(d)//HF/6-31 G(d) single point energies are indicated by dashed lines (see Figure 6 for structures).

Figure 4.8 Mechanism for the reaction of Ad=Ad + 2Br₂ (Pathway D). 180 Figure 4.9 Reaction pathway for the reaction of Ad=Ad + 28r₂ (Pathway D) at 181

HF/6-31 G(d) level of theory. For R⁰ and TS⁰ BJL YP/6- 31 G(d)//HF/6-31 G(d) single point energies are indicated by dashed lines (see Figure 8 for structures).

Chapter 5

Figure 5.1 Enthalpies of reaction (I) calculated at different levels of theory with 220 the standard 6-31 G( d,p) basis set.

Figure 5.2 Enthalpies of reaction (2) calculated at different levels of theory with 221

the standard 6-31 G(d,p) basis set.

Figure 5.3 Difference between cnthalpic of reaction (I) calculated at MP2 and 222 B3L YP levels of theory with G3MP2.

Figure 5.4 Difference between enthalpics of reaction (2) calculated at MP2 and 223 B3L YP levels of theory with G3MP2.

Figure 5.5 Enthalpy of reaction for Cll3Br + HCI - CH₃CI + HBr calculated at 224 different levels of theory and basis sets.

Figure 5.6 Enthalpy of reaction for iH3Br + HCI - SiH₃CI + IIBr calculated 225 at different levels of theory and ba is sets.

Figure 5.7 nthalpy of reaction for PH2Br + llC - PI-1₂C + IIBr calculated 226 at different levels of theory and basis sets.

List of Schemes

Chapter 4

Scheme 4.1 Reaction of Adamantylideneadamantane (I) with bromine in CCI~ ¹⁷⁰ forms adamantylideneadamantane bromonium ion with a Br

3·

counterion (2).

Scheme 4.2 Reaction of (E)-2,2,5,5-tetramethyl-3,4-diphenylhex-3-ene (3) with 171 bromine forms a CTC complex only.

Scheme 4.3 Tetraneopentylethylene (4) does not react with bromine 111 CCI

4 172

solution.

Chapter 6

Scheme 6.1 A possible mechanism for the reaction forming the largest side 253 product from the reaction between tetrabutylammonium cyanide and para-nitrobenzyl chloride in THF at 20°C.

A

Anhydr.

AO BJLYP BC Bz Bu4NCN

c

CI CIS CISD CTC D

OFT DMSO DZP Ea Et Fe 03

List of Abbreviation s and Sym bois

Angstrom, 1 o-Jo m Anhydrous

Atomic orbital

Becke3-Lee-Yang-Parr Binning-Curtiss

Benzyl

Tetrabutylammonium cyanide Concentration

Configuration interaction CI with single excitation

CI with single and double electronic excitations Charge transfer complex

Deuterium

Density functional theory Dimethyl sulphoxide

Double zeta polarized (basis set) Activation energy

Ethyl Frozen core Gaussian-n-theory

G3MP2 G3MP2B3 0383 GTF GTO H HF

ICR fRC

IUPAC K ks KIE KIEr LCAO LG MAD Me MO MPn MP2

Gaussian-n-theory Gaussian-n-theory Gaussian-n-thcory Gaussian type functions Gaussian type orital Planck's constant Hartree-Fock Intermediate

Ion cyclotron resonance Intrinsic reaction coordinate

International Union of Pure and Applied Chemistry Rate constant

Boltzmann's constant Kinetic isotope effect

Contribution to the isotope effect from tunnelling Linear combination of atomic orbitals

Leaving group

Mean absolute deviations Methyl

Molecular obital

Moller-Plessct perturbation theory of order n Second order Moller-Plesset perturbation theory

MS NMR Nu OTs PCM PES Ps PS QCl QCISD QCISD(T) R

r. c.

RS RHF SCF SCRF SN2

STO TDF THF TIF

Mass spectrometry

Nuclear magnetic resonance Nucleophile

Tosylate

Polarized continuum model Potential energy surface Picoseconds

Product state

Quadratic configuration interaction QCI including singles and doubles

QCISD with perturbative estimate for connected triples Gas constant

Reaction coordinate Reactant State

Restricted Hartree-Fock Self consistent field

Self consistent reactions field

Bimolecular nucleophilic substitution Slater type orbital

Temperature-dependent Factor Tetrahydrofuran

Temperature-independent factor

TS TST

UAO

UFF ZPE

v

E 1-1 N tor :f

Transition state Transition state theory United atom

Universal (united) force field Zero-point energy

Velocity

Dielectric constant Viscosity

Frequency

(Superscript) Relates to the transition state

List of Publications from This Th es is

I. Islam, S.M.; llollett, J. W; Poirier R. A. Computational Study of the Reactions or Sil bX (X = H, Cl, Br, I) with HCN, Journal of Physical Chemistry A, 2007, Ill, 526-540 (Chapter 2).

II. Islam, S. M.; Poirier. R. A. New insights into the bromination reaction for a serie of alkenes - A computational tudy, Journal of Physical Chemistry A, 2007, In Press (Chapter 3).

III. Islam, S. M.; Poirier. R. A. The addition reaction of adamantylideneadamantane with Br2 and 2Br2- A Computational tudy, Journal o( Physical Chemi.wy A., 2007, In Press (Chapter 4).

IV. Islam, S. M.; Huelin, S. D.; Dawe, M.; Poirier. R. A. A com pan son of the Standard 6-31 G and Binning-Curtiss Basis Sets for Third Row Elements, Journal of Chemical Theory and Computation, 2007, In Press (Chapter 5).

Y. Westaway, K. C.; Fang, Y.; MacMillar, S.; Matsson, 0.; Poirier. R. A.: Islam. . M. A new in ight into using chlorine leaving group and nuclcophile carbon kinetic isotope effects to determine substituent effects on the structure or SN2 transition states, Journal C?{Physical Chemistry A, 2007, Ill, 81 I 0-8120 (Chapter 6). Chapter 6 was mo tly written by Dr. Kenneth C. We taway, but I was actively

m olvcd in all the interpretations and analysis of the data and I produced all the necessary figures and tables. I have performed all the computations, while Dr.

Westaway's and Dr. Matsson's group performed the experiment .

VI. Westaway, K. C.; Fang, Y., MacMillar, .; Matsson, 0.; Poirier, R. .; I lam, . M. an incoming nucleophile carbon kinetic isotope effect be used to determine the transition state structure for different SN2 reactions?, submitted to Journal r~l Organic Chemistry (Chapter 7). hapter 7 was written jointly by Dr. Westaway, Dr. Poirier and I. I have performed all the calculations while the experimental work wa done by Dr. We taway's and Dr. Matsson·s group.

CHAPTER!

Introduction

1.1 Overall Goals and Objectives

Quantum mechanical computational chemistry is a fast growing research area.

Researchers use quantum chemistry calculations to predict and explain experimental results. Computational chemistry is used to study the structures, properties, and reactions of scientifically interesting systems. In principle, quantum mechanical computations can be very precise, because they can account for interactions between every electron and every proton in every atom. Experimental chemists have been investigating mechanisms of reactions for many years. However, it is very challenging to propose a reaction mechanism based on experimental observations because each step of the mechanism cannot be observed. The reactions are either very fast or cannot be carried out under normal conditions. The main purpose of this thesis is to investigate reaction mechanisms

computationally. Through computational methods, each step of a proposed reaction mechanism can be studied including the intermediates and transition states.

The reaction of bromine with alkenes is a well known organic reaction and many experimentalists have proposed possible mechanisms for the reaction under different reaction conditions. A survey of the literature reveals that very few theoretical studies^1-5 have been conducted for the bromination of alkenes. A computational study of the reactions of HCN with silanes and halosilanes has not been previously reported. These

reactions are very difficult to carry out experimentally, however it is very likely that such reactions may occur in interstellar space where both HCN and silanes are abundant.⁶ The reactions of silanes and halosilanes have also become a central focus in silicon chemistry for their usefulness in computers, semiconductors, polymer and glass industries.^7-10

Although there has been rapid advancement in theoretical chemistry over the past two decades, there are still many improvements to be made to increase the accuracy, applicability, and efficiency of computational methods. The potential for such advances is increased further by the continuing development of computer technology. The accuracy of the level of theory and basis sets used in predicting the properties of a system is crucial.

Therefore it is also necessary for computational chemists to compare the existing lower levels of theories, which are normally fast computationally, with higher level theories and with experiments. This will guide researchers in selecting the right level of theory and basis sets to use for their calculations. For ^3rd row elements, the Binning-Curtiss (BC6-

31 G) basis set¹¹ has been used as the 6-31 G basis set in most electronic structure packages (for example, Gaussian ¹² and GAMESS ¹³), although this basis set does not actually meet the definition of the standard 6-31 G basis set. Rassolov et al. ¹⁴ have constructed a standard 6-31 G basis set for the third row elements to utilize in G 3 theories.¹⁵ There has been no comparative study of the performance of the standard 6- 31 G and BC6-31 basis sets and researchers are unaware of the advantages and weaknesses of these basis sets in predicting the properties of a system.

The kinetic isotope effect (KIE) is normally used by experimentalists to characterize the transition state (TS) structure of a reaction. However, there are still many limitations associated with this experimental tool to determine ''the right" transition states.

It is possible to obtain the transition state structure computationally. To better understand the origin of KIEs, more and more experimental researchers are now comparing their results with computationally obtained TS structures and KIEs.¹⁶-19

However, more work needs to be done both experimentally and computationally to fmd and overcome the limitations associated with the KIE in predicting the TS structure successfully. This will allow one to use the KIE as a proper tool to characterize the TS structure for different reactions.

This thesis addresses each of the issues mentioned above. The broad objective of this thesis is given below in point form:

1. Identify the most plausible mechanism for the bromination reaction of alkenes by computational methods. The computational results will be compared to the data available from experiment.

2. Discover the likely mechanism for the reaction of HCN with SiH₃X, where X = H, Cl, Br, and I.

3. Determine the computationally least expensive level of theory that predicts reliable energies, by performing high level ab initio calculations and by comparing with experiment.

4. Determine the performance of the standard 6-31 G and the Binning-Curtiss (BC6- 31 G) basis sets by studying the geometries, frequencies and thermodynamic properties for molecules containing 3rd row elements.

5. Characterize the transition states of SN2 reactions by calculating the KlE and comparing them with the experimental results.

6. Find the unknown heats of formation of different compounds which have not been reported in the literature.

This thesis is arranged in manuscript format and contains a total of eight chapters including this introduction. Chapter - 1 presents the general introduction of the overall thesis and is divided into two sections. The first section in the introduction contains the overall goal and objective of this research including an outline of the thesis and the second section contains a brief background on the theory and methods used in this thesis.

All the chapters from 2 - 7 are arranged in manuscript format, each of which has its own

introduction, methodology, results & discussion, and conclusion summarizing the results.

Chapter - 2 presents a detailed study of the reactions of SiH3X (X = H, Cl, Br, I) with HCN. Chapter - 3 describes a detailed mechanistic and thermodynamic study of the bromination reaction for a series of alkenes in the gas phase and in solution. Chapter - 4 presents a detailed study of the reaction of adamantylideneadamantane with Br2 and 2Br2. Chapter - 5 is a comparative study of the performance of the standard 6-310 and Binning-Curtiss (BC6-31 G) basis sets for third row elements, Ga, Ge, As, Se, and Br.

Chapter - 6 presents a detailed study of chlorine leaving group and nucleophile carbon kinetic isotope effects to determine substituent effects on the structure of SN2 transition states. Chapter - 7 is also a detailed study of the use of nucleophile carbon kinetic isotope effects to determine the transition state structure for different SN2 reactions. Finally, a summary of the entire research work is presented in Chapter 8. Except for the last chapter, each chapter has its own references at the end.

1.2 Background

1.2.1 Transition State Theory

Transition state theory (TST) is a powerful theory for translating molecular structure and energetics into predictions of chemical reaction rates?⁰ In TST, the important criterion is that colliding molecules must have sufficient kinetic energy to overcome the activation energy barrier in order to react. The activation energy is the height of the potential energy barrier separating two minima on the potential energy

surface and the highest point is known as the transition state. The TS on such a surface is actually not a maximwn but a saddle point. According to transition state theory, the transition state, also known as the activated complex,²¹ is in equilibrium with the reactants,

A + B p

Reactants Transition State Product

According to the Arrhenius equation, the rate constant k is related to the activation energy Ea by the following equation,

(1) where, A is the pre-exponential factor, T represents the temperature and R is the gas constant.

The rate constant k of the elementary reaction is regulated by the difference in Gibbs' free energy between the reactant and the transition state, ~G# and can also be represented by the following equation,

k - kB T

-LlG"IRT

- - - e

h

⁽²⁾

where k₈, h, T and R represent the Boltzmann's constant, Planck's constant, the absolute temperature, and the gas constant, respectively. Therefore, Eq. (2) can be used to convert the experimentally obtained rate constant to the free energy of activation and vice versa.

compare the computationally obtained free energy of activation with the experimental free energy of activation. The computationally obtained free energies can also be converted to rate constants and compared to experiment.

The free energy of activation, ~G=F, can be represented by the enthalpy of activation, MI=F, and the entropy of activation, ~s=F, by equation (3) as given below,

(3) The structure of the transition state is often based on the structural features of the reactants and products. According to Hammond's postulate,²³ two states (reactant and transition state or transition state and product) of an elementary reaction having similar energies should only need a minor structural reorganization in order to interconvert. Thus, a reactant-like transition state structure is expected for an exothermic reaction, whereas an endothermic reaction will have a product-like transition state. There also exists a third case where the transition state occurs near the centre of a reaction coordinate and the reaction is known to be thermoneutral.

1.2.2 Kinetic Isotope Effect

The substitution of an atom in a molecule by one of its isotopes can alter the rate at which the molecule reacts. The ratio of the reaction rates is called the kinetic isotope effect (KIE). A KIE involving light (L) and heavy (H) isotopes is represented as

(4)

where

kL

is the rate constant for the molecule with the light isotope and

kH

is the rate

constant for the molecule with the heavy isotope. The KIE is normal when

..!sc.

> _{1.0 ,}

kH

inverse if

..!sc.

< 1.0 and there is no isotope effect if~= l.O.

kH kH

Isotopic substitution is a very useful technique to obtain information about the mechanistic sequence of a chemical reaction and the associated transition state structures.

The change of an isotope may cause the reaction rate to change in a number of ways, providing clues to the pathway of the reaction. For example, if the reacting molecule contains a C-H bond, then the reacting molecule will have a different reaction rate when protium is replaced by the heavier isotope deuterium, however, only if C-H is involved in the reaction. Because the zero-point energy of the C-D bond is lower than the C-H bond, a higher activation energy for C-D bond cleavage is required. The advantage of isotopic substitution is that this is the least disturbing structural change that can be effected in a molecule. Kinetic isotope effect is very useful to determine a transition state structure because it allows us to determine which bonds are forming or breaking in the transition state, and the amount of bond making and bond rupture that has occurred in the transition state. Primary KIEs result when bonds to the isotopically substituted atom are formed or broken in the rate determining step of a mechanism, while secondary KIEs result when the isotopically substituted atom influences the reaction rate but does not take part in the bond breaking/formation process. Figure 1.1 shows different types of KIEs from the

isotopic substitution in the reactants of an SN2 reaction. Some of these KIEs have been investigated in this thesis.

1.2.3 Origin of Kinetic Isotope Effect

Three situations can be considered for a reaction transition state structure, such as an early or reactant like TS, a late or product-like TS, and a symmetric or central TS.

Figures 1.2, 1.3 and 1.4 illustrate the primary KIE for a proton (H) or deuteron (D) transfer from a carbon (C) atom to another atom A and show the potential energy surface along with the zero point energies (ZPEs) for the stretch of the reactant state (RS) and the TS.

ZPE is one of the dominant factors for hydrogen/deuterium KIEs. In the RS, the C-H and C-D species have different energy due to their difference in ZPE. As can be observed from Figure 1.2, when the reaction is exothermic the TS is early, i.e. the C-H or C-D bond will only be slightly broken in the TS. The vibrational frequencies of the stretching mode in the TS will be affected by the mass of the H or D atom. Therefore, the difference in the TS energy for the C-H or C-D species will be almost the same in the transition structure as in the reactant. The difference in activation energies for reactions with a C-H or C-D will be very small and kH!k:o will be very close to unity. When the transition state is very late as for an endothermic reaction (Figure 1.3), the C-H or C-D bond will be almost broken and theA-Hand A-D bond fully developed. If the difference in ZPE for the A-H or A-D in the TS is very close to the ZPE for the C-H or C-D bond in

the RS, the activation energy for abstracting a deuterium is very similar to that for abstracting a hydrogen atom and the KIE will be very small. In a thermoneutral reaction, the TS structure is symmetrical and the frequency of the symmetric stretching vibration of the TS is almost independent of isotopic substitution, i.e the frequency of the vibration depends very little on the mass ofH or D. Therefore, the difference in isotopic ZPE in the RS will be fully expressed in the activation energies. As seen in Figure 1.4, the activation

energy for deuterium (Ea~ will be larger than the activation energy for hydrogen (EaH).

Thus, there should be a significant primary KIE. Westheimer and Melander^{3• 24} found that the maximum primary (1 °) KIE will be seen if the TS is symmetrical. The magnitude of such a maximum for a 1° KIE is estimated to be 7-10?^4,25 In the above analysis of the origin of kHiko, only stretching vibrations are considered and bending vibrations are completely ignored. But it is well known that the bending vibrations also play an important role in the magnitude of kHiko. ¹⁹,26

Researchers are still working on different types of kinetic isotope effects (KIEs) to better understand their origin. ^16-19,27

-28

1.2.4 Calculation of Kinetic Isotope effect

Using the assumptions of transition state theory, if molecular masses M, moments of inertia A, B and C, and isotopic vibrational frequencies vi are known for both reactants, R, and transition states, =/=, the kinetic isotope effect can be represented by the following equation, ²⁴'29

JN-6 sinh(J..LR I 2) JW'-7

sinh(J..L~

I 2)

X

I1

^{iL X}

I1

^{tH (5)}

i sinh(J..L~

I 2)

ⁱ sinh(J..L~

I 2)

where f..li = hv/ks T, h is Planck's constant, k₆ is the Boltzmann constant, T is absolute temperature and L and H denote the light and heavy isotopes of the reactants. By applying the Teller-Redlich rule³⁰, equation (5) can be represented by only the frequencies of the normal modes of vibration,

k (

~ J

^{3N-6 R}

^inh(

^R

^{I 2)}

_ L _ _- V L _{~ X}

n

^J..liH _R ^S _• ^J..liL _R

kH v H

ⁱ

J..LiL sinh(J..LiH I 2)

JN~-7 ^,,:

sinh(" :' 12)

X

n

^{r •L r•H}

i

J..L~1-1 sinh(J..L~L I 2)

⁽⁶⁾

The above equation neglects the tunneling contribution to isotope effects. The significance of tunneling is that a light particle such as proton can cross an energy barrier

with less energy than the activation energy, whereas a heavy particle such as deuterium cannot do so as easily. The tunneling effect which may be very important can be accounted for by the Wigner correction³¹, where the first two terms of the following expansion series are calculated,

At 298K,

2

Qt = 1+ ~~ +···

hvi 1.44

' '1 =-~--xv. ~o.004828xv.

,... k T T

^{I I}

B

(7)

(8)

where, vi is the imaginary frequency of the transition state. The tunneling contribution to the K1E is obtained from the ratio of QtLIQtH· Tunneling is mainly important for the reactions with narrow reaction barriers. Tunneling is also favoured by low temperature and light nuclei. Therefore a kinetic isotope effect can be expressed as the product of three factors, KlET, TIF and TDF, according to equation (9),^24,29

(9)

TIF TDF

KIET is the contribution from tunneling to the total KIE. The temperature independent factor (TIF) is the ratio of the imaginary frequencies for the isotopic transition states. The ratio of the total isotopic vibrational frequencies in the transition states to that of the reactant constitutes the temperature-dependent factor (TDF). TDF can also be represented by the products of the normal mode vibrational frequencies (VP), excited vibrational modes (EXC) and zero point energies of the vibrational modes (ZPE).2⁹ KlET is always normal as the lighter isotope is affected more by tunnelling than the heavier isotope. KIET may become very important when the magnitude of the KIE is small. Since the species containing the lighter isotope has the higher frequency, TIF will always be greater than unity. The change in the vibrational frequencies on going from the reactant state to the transition state determines the TDF. Therefore TDF will be inverse if the binding to the isotopically labeled atom is increased. For example, bond formation to a nucleophile

results in increased vibrational energy in the TS compared to the reactant. The TDF will be increasingly inverse with the degree of bond formation in the TS. Therefore, the magnitude of an incoming group KlE will be normal if there is a small amount of bond formation in the TS whereas it will be inverse if bond formation is far advanced in the TS. ³²-34 This type of KIE can be very small and close to unity as the TIF and TDF have the opposite contributions to the total KIE.

1.2.5 The Molecular Hamiltonian

Finding an approximate solution of the Schrodinger equation is the main focus area of the computational chemist. The time-independent, non-relativistic Schrodinger equation is written as³⁵'36

H\f'

ⁱ

(r,R.)

= Ei

\f'

ⁱ

(r,R.)

(10) where,

r

^and

R

are the electronic and nuclear coordinates, respectively. 'f', is the wave function of the i'th state of the system. 'f', depends on 4N variables, three spatial variables (coordinates) and one spin variable for each of theN electrons, along with 3M spatial coordinates of the nuclei. Ei represents the total energy of the system.

H

^{is the}

Hamiltonian operator (or simply, the Hamiltonian) for a molecular system consisting of M nuclei and N electrons and contains all the terms that contribute to the total energy of the system. The Hamiltonian,

H ,

can be written in atomic units as:

(11)

The first two terms in the equation (11) describe the kinetic energy of the electrons and nuclei, respectively. The third term represents the attractive electrostatic interaction between the nuclei and the electrons (Coulomb attraction). The last two terms represent the repulsive potential due to the electron-electron and nucleus-nucleus interaction, respectively. V~ and V~ are the Laplacian operators resulting from the differentiation with respect to the coordinates of the electron and nucleus, respectively. ZA and ZB are the nuclear charge of nucleus A and B, respectively and the distances between the

1.2.6 Born-Oppenheimer Approximation

Because the masses of the nuclei are several orders of magnitude greater than that of an electron, they move much slowly and can be considered stationary and separable while computing the electronic energies. This is the basis of the famous Born- Oppenheimer approximation. ³⁷ Therefore, the Schrodinger equation can be written in terms of the electronic Hamiltonian operator as

"'

Helec 'I' elec - Eelec 'I' elec (12)

where the electronic Hamiltonian is the reduced form of the complete Hamiltonian given in Eq. (11),

(13)

'¥ elec depends on the electron coordinates and parametrically on the nuclear coordinates.

Adding the constant nuclear repulsion term gives the total energy as

(14) Therefore the electronic and nuclear parts of the Schrodinger equation can be solved separately by introducing the parametric dependence of the total energy on the nuclear coordinates. In more general terms, for a given set of nuclear configurations, the electronic Schrodinger equation is solved to find the total energy, allowing the energy to be studied as a function of the nuclear coordinates. This provides the potential energy surface (PES) of the system.

1.2. 7 The Hartree-Fock Approximation

It is not possible to solve the electronic Schrodinger equation exactly with more than two interacting particles and therefore further simplification is required. One such simplification is the use of an approximate wave function, such as a single Slater determinant. This is known as the Hartree-Fock approximation. The formalism is outlined by Szabo and Ostlund³⁵ in detail. The HF energy can be represented in chemist's notation as

N/2 N/2 N/2

EHF =

j'l'•fr¥d't

⁼

('~'jHj'l')

⁼

2L( ^a!hla )+ LL2( aalbb )- ( ablba)

a a b

N/2 N/2 N/2

=

2Lhaa + LL2Jab - Kab ⁽¹⁵⁾

a a b

where

H

is the full electronic Hamiltonian, N is the number of electrons, 'I' is the wave function and spatial orbitals are denoted by a and b. The first term in Eq. (15) is the one- electron integral as shown in Eq. (16),

(16)

The above equation defines the contribution from the kinetic energy of an electron and the attractive electrostatic interaction between electron and nucleus.

The second term in Eq. (15) is defined as,

Eq. ( 17) and Eq. ( 18) represent the Coulomb integrals, Jab and the exchange integrals, Ka!,, respectively, between two electrons.

According to the variation principle, the best wave function is the one which gives the lowest possible energy (E0), i.e,

(19)

Minimizing Eo with respect to the spatial orbitals leads to the Hartree-Fock equations,

i=1,2, ... N (20)

A

where f is called Fock operator which acts on the molecular orbital 'I'; .

A

The f is an effective one-electron operator defined as:

f(i) = _ _!_

v~- f ^zA

^{+vHF (i) = h(i) +vHF (i)}

2 A liA

(21)

Where, h(i) is the core-Hamiltonian operator and vHF (i) is the effective one-electron potential operator called the Hartree-Fock potential which represents the average repulsive potential experienced by the i 'th electron due to the presence of all the other electrons.

1.2.8 The Basis Set Expansion

It is necessary to specify the form of spatial orbitals 'I'; (r) to carry out calculations on molecular systems. The orbitals may be expressed as a linear combination of a set of basis functions { cJ>Il, J.1 = 1, 2, .. · K} :

K

\If; (r) =

L cjli<~>jl

^{(r) (22)}

jl=l

where <l>ll (r) are basis functions, ^Clli are the molecular orbital expansion coefficients.

Roothaan's equation can be written as a single matrix equation,³⁵• 3 6

FC=cSC

(23)

where F is the Fock matrix where each element can be written as:

(24) S is the overlap matrix where each element can be written as:

(25)